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Stochastic Partial Differential Equations

In: Stochastic Systems

Author

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  • Mircea Grigoriu

    (Mircea Grigoriu Cornell University)

Abstract

A brief discussion on the relevance of stochastic partial differential equations (SPDEs) in Sect. 9.1 is followed by a review of the type of SPDEs studied in the mathematical literature (Sect. 9.2). Section 9.3 shows that SPDEs can be solved by the methods in Chaps. 7 and 8 via time and space discretization. Section 9.4 deals with SPDEs that are frequently encountered in applications. The focus is on stochastic elliptic partial differential equations. We review general concepts on this class of equations (Sect. 9.4.1), and present solutions by Monte Carlo (Sect. 9.4.4), stochastic reduced order models (Sect. 9.4.5), stochastic Galerkin (Sect. 9.4.8), and stochastic collocation (Sect. 9.4.9) methods. The last section of the chapter considers SDEs whose coefficients have small uncertainty, and presents methods for solving these equations by Taylor (Sect. 9.5.1), perturbation (Sect. 9.5.2), and Neumann series (Sect. 9.5.3) representations.

Suggested Citation

  • Mircea Grigoriu, 2012. "Stochastic Partial Differential Equations," Springer Series in Reliability Engineering, in: Stochastic Systems, edition 127, chapter 0, pages 379-454, Springer.
    • Handle: RePEc:spr:ssrchp:978-1-4471-2327-9_9
    DOI: 10.1007/978-1-4471-2327-9_9
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